Exploding Dots

1.3 A Peek at a Weird Machine

Lesson materials located below the video overview.

Let’s get quirky! Here are two first attempts at weirdness:


What do you think of a \(1 \leftarrow 1\) machine?

What happens if you put in a single dot? Is a \(1 \leftarrow 1\) machine interesting? Helpful?


What do you think of a \(2 \leftarrow 1\) machine?

What happens if you put in a single dot?

What do you think of the utility of a \(2 \leftarrow 1\) machine?


Okay … How about this then?


Consider the \(2 \leftarrow 3\) machine.

This machine replaces three dots in one box with two dots one place to their left.


This machine seems to do interesting things. For example, placing ten dots into the machine produces the following result:


\(10  \rightarrow 2101\)

(Check this!)

Here are the \(2 \leftarrow 3\) codes for the first fifteen numbers:


Hmm. Do these mean anything?


Some questions:

Does it make sense that only the digits \(0\), \(1\), and \(2\) appear in these codes?
Do all codes after then number one begin with \(2\)? Begin with \(21\) (from six onwards)?
Does it make sense that the final digits of these codes cycle \(1, 2, 0, 1, 2, 0, 1, 2, 0, ….\)?
Can one do arithmetic in this weird system? For example, here is what \(6+5\) looks like. Is the answer indeed eleven?


(Actually, maybe the real answer is \(402\) or better \(2102\). Do you see what I am thinking? Is this the proper code for eleven?)


But the real question is:

What are these codes? What are we doing representing numbers this way?

Are these codes for numbers written in base three? In base two? In some other base?

There are a lot of shockers to be had about this \(2 \leftarrow 3\) machine. We’ll explore this machine in detail in section 3.1 of this course, and present a number of open research questions about it.

For now, just play with the \(2 \leftarrow 3\) machine and get a feel for it. Perhaps write out the codes for all the numbers up to fifty and look for structure and patterns. And think about the question: Are we writing numbers in some base? If so, what base?  More on this, for sure, later!

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